Today we will prove that the area of a circle with radius r equals pi r squared. Our method involves dividing the circle into many small triangular sectors and rearranging them to form a rectangle.
First, we divide the circle into equal triangular sectors. Each sector has an arc as its base and the radius as its height. The central angle of each sector is 2 pi divided by n, where n is the number of sectors. As n increases, the sectors become thinner and more triangle-like.
Next, we cut out the sectors and rearrange them side by side, alternating their orientation. One sector points up, the next points down, and so on. This arrangement forms an approximate parallelogram. The height of this shape is the radius r, and the base length is half the circumference, which is pi r.
Today we will prove that the area of a circle is pi r squared using a clever geometric approach. We start by dividing the circle into many triangular sectors, like slices of a pie. Each sector has its vertex at the center of the circle.
Let's examine one sector more closely. Each sector is approximately a triangle. The base of this triangle is the arc length, and the height is the radius r. When we divide the circle into n equal sectors, each arc length becomes 2 pi r divided by n.
Now we rearrange all the sectors by placing them alternately - some pointing up and others pointing down. This creates a shape that resembles a rectangle with zigzag edges. The total width is 2 pi r, which is the circumference of the circle, and the height is r, the radius.
As the number of sectors approaches infinity, the curved edges become perfectly straight, and the shape becomes a perfect rectangle. The height is r and the base is pi r. The area of this rectangle is base times height, which equals pi r times r, giving us pi r squared. This proves that the area of a circle is pi r squared.
To summarize what we have learned: We divided the circle into triangular sectors and rearranged them into a rectangular shape. As the number of sectors increases, the edges become straight. The resulting rectangle has area pi r squared, which proves the circle area formula.
To summarize what we have learned: We divided the circle into triangular sectors and rearranged them into a rectangular shape. As the number of sectors increases, the edges become straight. The resulting rectangle has area pi r squared, which proves the circle area formula.